Need some advice: Can you name a child a 3rd after his great grandfather?

My wife and I are going back and forth on this. We're having a son, and neither of us had many strong father figures in our lives -- except my grandfather, James H Jr.. Apparently I was almost named James H III myself.

So I never thought it was weird that I would one day name a son James H the 3rd, to continue from my Grandfather. But when I threw the suggestion out there, my wife is adamant that we "Can't" do that, because I am Joel G, not James H, so at most all I can have is a Joel G Jr, which I don't care for.

Google seems to think I also would be breaking convention, but is it really that outlandish a suggestion? Am I setting my son up for failure when he has to explain that he is a 3rd of his name, but he doesn't share his father's name?

Thank you.

PS: I do have a compromise offer with my wife that I am open to naming him Joel Twopointoh...